Some excerpts from link:
France introduced Martinique to absinthe, an alcohol the island continued to produce despite France's prohibition by 1915. As a result, Latamie grew up with absinthe regularly brewed in homes throughout Martinique, and recalls that as a child he would sniff the absinthe perfume kept in his grandmother's cabinet.
The installation will also feature a short film by Myrtha Richards-Marie-Joseph, who is 2012 made "L'Absinthe de Monsieur Gentil" (Mr. Gentil's Absinthe), which follows a man in Martinique as he distills absinthe in his home.
I would guess that if any local absinthe was made in Martinique, it would have started with a rum/ sugar cane base?